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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Let <span class="process-math">\(\alpha=1\)</span> and <span class="process-math">\(L=1\text{.}\)</span> If <span class="process-math">\(f(x)=10\sin \pi x\text{,}\)</span> then we have <span class="process-math">\(C_1 =10\)</span> and all other <span class="process-math">\(C_n = 0\text{,}\)</span> the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-\pi^2t}\sin \pi x.
\end{equation*}
</div>
<p class="continuation">If now let <span class="process-math">\(f(x) = 10\sin 3\pi x\text{,}\)</span> then <span class="process-math">\(C_3 = 10\)</span> and all other <span class="process-math">\(C_n = 0\text{,}\)</span> and the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-9\pi^2t}\sin 3\pi x.
\end{equation*}
</div>
<p class="continuation">If the initial temperature is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(x)=10\sin \pi x + 10\sin 3\pi x,
\end{equation*}
</div>
<p class="continuation">the solution would be</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-\pi^2t}\sin \pi x +10e^{-9\pi^2t}\sin 3\pi x.
\end{equation*}
</div>
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